Class 9 Mathematics - Coordinate Geometry Notes & Solutions

Chapter 3: Coordinate Geometry (NCERT Solutions)

Exercise 3.1

Q1. How will you describe the position of a table lamp on your study table to another person?

To describe the position of a table lamp on the study table, we assume the table to be a plane surface and the lamp to be a point on the plane. Then we follow these steps:

1. Choose two adjacent edges of the table as our axes - the longer edge as the horizontal axis (x-axis) and the shorter edge as the vertical axis (y-axis).
2. Measure the perpendicular distance of the lamp from the shorter edge. Let it be 25 cm (this is the x-coordinate).
3. Measure the perpendicular distance of the lamp from the longer edge. Let it be 30 cm (this is the y-coordinate).
4. So, the position of the lamp can be described by the ordered pair (25, 30). Alternatively, depending on the choice of axes, it can also be correctly described as (30, 25).

Q2. (Street Plan): A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction. All the other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.
There are many cross-streets in your model. A particular cross-street is made by two streets, one running in the North-South direction and another in the East-West direction. Each cross street is referred to in the following manner: If the 2nd street running in the North-South direction and 5th in the East-West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:
(i) how many cross-streets can be referred to as (4, 3).
(ii) how many cross-streets can be referred to as (3, 4).

Solution:
(i) A unique point represents the crossing of the 4th street in the North-South direction and the 3rd street in the East-West direction.
Therefore, only 1 cross-street can be referred to as (4, 3) because two distinct intersecting lines have only one unique point of intersection.

(ii) Similarly, a unique point is defined when the 3rd street running North-South intersects the 4th street running East-West.
Therefore, only 1 cross-street can be referred to as (3, 4).

Exercise 3.2

Q1. Write the answer of each of the following questions:
(i) What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?
(ii) What is the name of each part of the plane formed by these two lines?
(iii) Write the name of the point where these two lines intersect.

(i) The horizontal line taking part in plotting a point is called the x-axis, and the vertical line is called the y-axis.

(ii) The two axes coordinate plane into four regions. Each part is called a Quadrant.

(iii) The point where the x-axis and y-axis intersect is called the Origin (0, 0).

Q2. See Fig. 3.14, and write the following:
(i) The coordinates of B.
(ii) The coordinates of C.
(iii) The point identified by the coordinates (-3, -5).
(iv) The point identified by the coordinates (2, -4).
(v) The abscissa of the point D.
(vi) The ordinate of the point H.
(vii) The coordinates of the point L.
(viii) The coordinates of the point M.

(Note: Based on the standard NCERT textbook figure 3.14):

(i) The coordinates of B are (-5, 2). (It is 5 units left and 2 units up).

(ii) The coordinates of C are (5, -5). (It is 5 units right and 5 units down).

(iii) The point identified by the coordinates (-3, -5) is E.

(iv) The point identified by the coordinates (2, -4) is G.

(v) The abscissa (x-coordinate) of the point D(6, 2) is 6.

(vi) The ordinate (y-coordinate) of the point H(-5, -3) is -3.

(vii) The coordinates of the point L, which lies on the y-axis, are (0, 5).

(viii) The coordinates of the point M, which lies on the x-axis, are (-3, 0).

Exercise 3.3

Q1. In which quadrant or on which axis do each of the points (- 2, 4), (3, - 1), (- 1, 0), (1, 2) and (- 3, - 5) lie? Verify your answer by locating them on the Cartesian plane.

(-2, 4): The x-coordinate is negative and the y-coordinate is positive (-, +). So, it lies in the II Quadrant.

(3, -1): The x-coordinate is positive and the y-coordinate is negative (+, -). So, it lies in the IV Quadrant.

(-1, 0): The y-coordinate is zero, meaning it lies directly on the horizontal line. Since the x-coordinate is negative, it lies on the Negative X-axis.

(1, 2): Both the x-coordinate and y-coordinate are positive (+, +). So, it lies in the I Quadrant.

(-3, -5): Both the x-coordinate and y-coordinate are negative (-, -). So, it lies in the III Quadrant.

Q2. Plot the points (x, y) given in the following table on the plane, choosing suitable units of distance on the axes.

x	-2	-1	0	1	3
y	8	7	-1.25	3	-1

To plot these points, we draw the X-axis and Y-axis perpendicular to each other. Taking an arbitrary unit (say 1 cm = 1 unit), we plot:

- Point A(-2, 8) is 2 units to the left of the y-axis, and 8 units up from the x-axis. (II Quadrant)
- Point B(-1, 7) is 1 unit to the left of the y-axis, and 7 units up from the x-axis. (II Quadrant)
- Point C(0, -1.25) is on the negative y-axis. It is located 1.25 units down from the origin.
- Point D(1, 3) is 1 unit to the right of the y-axis, and 3 units up from the x-axis. (I Quadrant)
- Point E(3, -1) is 3 units to the right of the y-axis, and 1 unit down from the x-axis. (IV Quadrant)

(Draw an actual graph in your notebook locating these coordinates based on the descriptions provided.)

Class 9 Mathematics - Coordinate Geometry Numericals & Practice

Chapter 3: Coordinate Geometry (Practice Questions)

RD Sharma / Extra Practice

Q1. Write the names of the quadrants in which the following points lie: A(2, 5), B(-3, 4), C(-2, -6), D(7, -3).

A(2, 5) ⇒ (+,+) ⇒ Quadrant I
B(-3, 4) ⇒ (-,+) ⇒ Quadrant II
C(-2, -6) ⇒ (-,-) ⇒ Quadrant III
D(7, -3) ⇒ (+,-) ⇒ Quadrant IV

Q2. Plot the points (0, -4), (3, 0), and (-4, 0) and name the axis on which they lie.

(0, -4): Since x-coordinate is 0, it lies on the Negative Y-axis.
(3, 0): Since y-coordinate is 0, it lies on the Positive X-axis.
(-4, 0): Since y-coordinate is 0, it lies on the Negative X-axis.

Q3. Without plotting the points on a graph, indicate the quadrant in which they will lie, if:
(i) Ordinate is 5 and abscissa is -3.
(ii) Abscissa is -5 and ordinate is -3.
(iii) Abscissa is -5 and ordinate is 3.
(iv) Ordinate is 5 and abscissa is 3.

(i) (-3, 5): -,+ ⇒ Quadrant II
(ii) (-5, -3): -,- ⇒ Quadrant III
(iii) (-5, 3): -,+ ⇒ Quadrant II
(iv) (3, 5): +,+ ⇒ Quadrant I

Q4. Write the perpendicular distance of the point P(4, 3) from the x-axis and y-axis.

Perpendicular distance from the x-axis = |y-coordinate| = 3 units.
Perpendicular distance from the y-axis = |x-coordinate| = 4 units.

Q5. If the coordinates of two points are P(-5, 3) and Q(8, -9), then find (Abscissa of Q) - (Abscissa of P).

Abscissa of Q = 8
Abscissa of P = -5
Difference = 8 - (-5) = 8 + 5 = 13.

Q6. Find the coordinates of the point:
- whose ordinate is -4 and which lies on the y-axis.
- whose abscissa is 5 and which lies on the x-axis.

If a point lies on the y-axis, its x-coordinate is 0. So, coordinates are (0, -4).
If a point lies on the x-axis, its y-coordinate is 0. So, coordinates are (5, 0).

Q7. In which quadrant or on which axis does the point (a, b) lie, if a < 0 and b> 0?

If a < 0 (negative x-coordinate) and b> 0 (positive y-coordinate), the point (a, b) is of the form (-,+).
Therefore, it lies in the II Quadrant.

Q8. A point lies on both the x-axis and the y-axis. What are its coordinates?

The only point simultaneously lying on both axes is the point of their intersection.
Therefore, the coordinates are (0, 0), which is the Origin.

Q9. What is the perpendicular distance of the point (-6, 5) from the y-axis?

The perpendicular distance from the y-axis equals the absolute value of the x-coordinate.
Distance = |-6| = 6 units.

Q10. The points (other than origin) for which abscissa is equal to the ordinate will lie in which quadrants?

If Abscissa = Ordinate ⇒ x = y.
This means the point is either (+,+) or (-,-).
Therefore, they lie in Quadrant I and Quadrant III.

Q11. The points for which ordinate is equal to negative of abscissa will lie in which quadrants?

If Ordinate = -Abscissa ⇒ y = -x.
This means if x is positive, y is negative (+,-). If x is negative, y is positive (-,+).
Therefore, they lie in Quadrant II and Quadrant IV.

Q12. What are the coordinates of the mirror image of a point (-4, 5) in the x-axis?

If a point (x, y) is reflected across the x-axis, its x-coordinate remains identical, but the y-coordinate changes sign. ⇒ (x, -y).
So, the mirror image of (-4, 5) is (-4, -5).

Q13. What are the coordinates of the mirror image of a point (3, -2) in the y-axis?

If a point (x, y) is reflected across the y-axis, its y-coordinate remains identical, but the x-coordinate changes sign. ⇒ (-x, y).
So, the mirror image of (3, -2) is (-3, -2).

Q14. State the quadrant in which point P(0, 4) lies.

Point P(0, 4) has an x-coordinate of 0. Therefore, it does not lie in any quadrant. It lies on the Y-axis.

Q15. Write the coordinates of the point whose abscissa is the solution of the equation: 2x - 6 = 0 and whose ordinate is the solution of the equation: 3y + 9 = 0.

Solving 2x - 6 = 0: 2x = 6 ⇒ x = 3 (Abscissa).
Solving 3y + 9 = 0: 3y = -9 ⇒ y = -3 (Ordinate).
The coordinates are (3, -3).

Q16. Find the area of the triangle formed by the points A(2, 0), B(6, 0), and C(4, 6) on the graph.

Points A(2,0) and B(6,0) lie on the x-axis.
The base of the triangle is the distance between A and B.
Base length = 6 - 2 = 4 units.
Height of triangle corresponds to the perpendicular distance from C(4, 6) to the x-axis, which is the y-coordinate.
Height = 6 units.
Area = ½ × base × height = ½ × 4 × 6 = 12 sq. units.

Q17. A square is placed such that its vertices are exactly midway between the axes. If the diagonals of the square are 10 units long and intersect at the origin, find the coordinates of its vertices.

Since the diagonals are 10 units long and intersect at the origin, the distance from origin to each vertex is 10/2 = 5 units.
The vertices run along the axes.
The coordinates will be at an equal 5 unit length across the X and Y axes in each direction.
Coordinates: (5, 0), (0, 5), (-5, 0), (0, -5).

Q18. Point A(-3, 2) is translated by 5 units to the right and 4 units downwards. Find the new coordinates.

5 units right means adding 5 to the x-coordinate: -3 + 5 = 2.
4 units downward means subtracting 4 from the y-coordinate: 2 - 4 = -2.
The new coordinates are (2, -2).

Q19. Three vertices of a rectangle are (3, 2), (-4, 2) and (-4, 5). Find the coordinates of the fourth vertex without graphing.

The first two points have the same y-coordinate (2). This horizontal side goes from x = -4 to x = 3.
The vertical side runs from (-4, 2) to (-4, 5).
To complete the rectangle, the 4th vertex must vertically align with x = 3 (from the first point) and horizontally align with y = 5 (from the third point).
Therefore, the fourth vertex is (3, 5).

Q20. P is a point on the x-axis at a distance of 3 units from the y-axis to its left. Let Q and R act as its mirror reflections over the Y-axis and X-axis respectively. What are their coordinates?

Point P lies on the x-axis, 3 units to the left: P(-3, 0).
Reflection of P across the y-axis means turning x-coordinate positive: Q(3, 0).
Reflection of P across the x-axis means negating y-coordinate: Since y is 0, it doesn't change: R(-3, 0).
So Q is (3, 0) and R is (-3, 0).

Class 9 Mathematics - Coordinate Geometry Formulas & Summary

Chapter 3: Coordinate Geometry (Concepts & Formulas)

1. The Cartesian System

Cartesian Plane: The coordinate axes (X-axis and Y-axis) divide the plane into four separate regions called quadrants. The plane consists of the axes and these quadrants. We call the plane the Cartesian plane, or the coordinate plane, or the xy-plane.
Coordinate Axes: The two perpendicular number lines drawn in the plane are known as the coordinate axes.
- The horizontal line is the X-axis (typically labeled X'OX).
- The vertical line is the Y-axis (typically labeled Y'OY).
Origin: The point of intersection of the coordinate axes is called the origin, denoted usually by O (0, 0).

2. Coordinates of a Point

The location of a point is given by an ordered pair (x, y).
Abscissa (x-coordinate): The x-coordinate of a point is its perpendicular distance from the y-axis measured along the x-axis (positive right of origin, negative left of origin).
Ordinate (y-coordinate): The y-coordinate of a point is its perpendicular distance from the x-axis measured along the y-axis (positive above origin, negative below origin).

3. Quadrants Rule

The axes divide the plane into four quadrants, going in an anti-clockwise direction starting from OX and OY:

Quadrant I: Both X and Y are positive ⇒ (+, +)
Quadrant II: X is negative, Y is positive ⇒ (-, +)
Quadrant III: Both X and Y are negative ⇒ (-, -)
Quadrant IV: X is positive, Y is negative ⇒ (+, -)

4. Points on the Axes

Any point lying on the X-axis has its y-coordinate equal to zero. Its form is (x, 0).
Any point lying on the Y-axis has its x-coordinate equal to zero. Its form is (0, y).
The intersection of both axes is strictly (0, 0).

5. Important Review Rules

The coordinates of a point are always written as an ordered pair inside round brackets, with the x-coordinate always written first. e.g. (x, y).
(x, y) ≠ (y, x) unless x = y. Position is entirely unique to the ordered pair sequence.
Mirror image reflections:
- Over X-axis: (x, y) ⇒ (x, -y)
- Over Y-axis: (x, y) ⇒ (-x, y)
- Over Origin: (x, y) ⇒ (-x, -y)

📱 Practice MCQs for this topic inside our App